Broadly speaking, the project addresses questions on the interface of Poisson geometry, representation theory, functional analysis, and quantum topology. The first field, Poisson geometry, can be thought of as a mathematical language for classical mechanics. The next two, representation theory and functional analysis, constitute the language for quantum mechanics. The last one, quantum topology, provides mathematical tools for topological quantum field theories (TQFT. Finally, the combination of all 4 of them, gives a broad machinery applicable, for example, to the study of gauge theories. More specifically, my research in this rather broad collection of mathematical areas revolves around the so-called, cluster algebras and cluster varieties. The latter may be understood as a yet another mathematical tool that provides an infinite number of extremely convenient coordinate systems (exponentiated Darboux coordinates), in which a large number of non-trivial calculations get significantly simplified.
The types of mathematical problems this project addresses are essentially of the following 3 kinds. The first one concerns constructions when certain explicitly computable algebraic data (such as a representation of an algebra) is attached to a geometric object (such as a surface), in a way that distinguishes these objects up to their topology. In this way we can speak of topological invariants of surfaces. The second kind are problems in representation theory and Poisson geometry, which either relate representations of different algebras, thus building connections between the corresponding physical systems, or relate constructions in quantum algebra to their quasi-classical analogues, thus bridging the quantum and classical systems. Finally, the third type of problems I work on is the search of cluster structures on classical or quantum objects, in order to apply the powerful machinery of cluster coordinates to questions above.
My proposal has two main objectives. The first one is to establish a proof of the celebrated Modular Functor Conjecture in higher Teichmüller theory. In broad terms, the conjecture seeks to construct a partial data of a 4-dimensional TQFT, providing topological invariants of low-dimensional manifolds. However, the construction utilizes functional analytic rather than algebraic representation theory, thus bringing us one step closer to models of actual interest to broader physics community. The second major objective is to establish cluster structure on the so-called Coulomb branches of the so-called 4d N=2 quiver gauge theories. One may think that the theory takes a group and a surface as an input, and produces a Poisson manifold as one of its outputs, the so-called Coulumb branch. Those have been of great interest for physics community for several decades and has been finally given a mathematical definition by Braverman, Finkelberg, and Nakajima less than a decade ago. It was conjectured by Gaiotto that their Coulomb branches possess cluster structure and the second objective is to settle that conjecture.
